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Abstract

This paper provides a new approach for the design of small module gear with curve configuration, and studies the meshing temperature rise of the generated tooth surface under dry running condition. To predict the meshing temperature of small module gear with curve configuration, the normal load and motion velocity of two contact points are analyzed based on vector mapping method, and the contact stress and contact deformation are obtained based on Hertzian contact theory. The frictional heat flux at different speed and torque is calculated as the heat source, and the convective heat transfer coefficient of tooth surfaces are calculated as the heat dissipation condition. Finite element model of temperature field of small module gear with curve configuration is established, and the meshing temperature field is obtained. Variation trend of the maximum steady-state temperature is in good agreement with the experimental test results. Simulation results of flash temperature also agree well with Blok theory. The results show the rotational speed and the torque are high, the meshing temperature rise of the gear pair will be high. The maximum temperature is distributed in the middle area of single tooth on the first contact trace of the gear.

Keywords

Small module gear with curve configuration meshing model contact analysis thermal analysis temperature field

Introduction

Small modulus gears (m_n ≤ 1) are widely used in automotive electronics, smart home, and other scenarios, and are very important industrial basic parts. In most working scenarios, the small module gear pair moves through grease lubrication or dry friction, and the grease will be reduced or lost due to factors such as high temperature, impurities, or additive failure during use.¹ Due to its size limitation and working conditions, the heat dissipation of the tooth surface of the small module gear is poor, which easily leads to the failure of the meshing temperature rise gear. Therefore, the design and meshing characteristics analysis of high-performance small module gear pair is meaningful. Sun et al.² studied the failure problems of small module gears such as tooth root fracture and tooth surface fatigue pitting caused by dynamic wear. Schultheiss et al.^3,4 studied the wear characteristics of small module gears under grease lubrication conditions. Problems such as friction and wear, fatigue pitting, and tooth surface scuffing failure greatly limit the working life of small module gears. Dobler et al.⁵ studied how to improve the bending strength and pitting load resistance of small module gears with involute tooth profile.

Some scholars have also studied the principle of gear meshing temperature rise. Blok⁶ first proposed the theory of flash temperature in 1963. Černe et al.^7–9 proposed an improved flash temperature prediction model, which can more accurately predict the heat loss caused by friction-induced temperature rise. The maximum meshing temperature generated by the friction heat of the gear transmission is composed of the steady-state temperature of the tooth body and the flash temperature of the tooth surface.¹⁰ Mao¹¹ analyzed the flash temperature of gear and the heat distribution between teeth in detail. The intensity distribution and velocity of heat source vary with the area of grid division, and the finite difference method is used for numerical approximation. At present, the mainstream research methods for the meshing temperature rise of gear pairs are finite element thermal grid simulation analysis^12–14 and experimental research.^15–18 There have been many studies on the prediction of the meshing temperature rise of involute gears with linear contact. Li and Kahraman¹⁹ established the contact wear model of spur gear, and proposed that the radius of curvature, surface velocity, and normal force of gear meshing at the meshing contact line change with time. Koffi et al.²⁰ established a theoretical model for evaluating the temperature rise of gear meshing. Roda-Casanova and Sanchez-Marin²¹ simplified the three-dimensional spur gear model into a two-dimensional plane model, and predicted the gear meshing temperature field based on finite element simulation, which improved the calculation efficiency. Terauchi and Mori²² studied the effects of gear speed, tooth surface load, and lubricant on wheel temperature rise. Fernandes et al.²³ used the finite element thermal mesh model to predict the body temperature and flash heat of the straight-tooth involute polymer cylindrical gear. Based on the finite element analysis, Li et al.²⁴ concluded that the temperature distribution of the spur gear is symmetrically distributed along the tooth width direction, while the temperature distribution of the helical gear is asymmetrically distributed along the tooth width direction.

In addition to the involute tooth profile of the line contact gear pair, the meshing contact state for the point contact of the meshing temperature rise prediction also has some research. Cao et al.^25,26 studied the lubrication characteristics, friction coefficient, flash temperature, and fatigue life characteristics of spiral bevel gears under conventional working conditions through the mixed EHL model of spiral bevel gears. Pu et al.²⁷ further studied the mixed EHL model of spiral bevel gears, and numerically simulated the friction coefficient under different entrainment angles and entrainment speeds. Morales-Espejel²⁸ studied the problem of sharp temperature rise in rolling contact based on creep fatigue theory and mixed EHL model. Wang et al.²⁹ reduced the temperature rise of gear meshing by optimizing the parameters of spiral bevel gears. At present, there is no research on the meshing temperature rise of gear with curve configuration.

In this paper, a new type of small module gear with curve configuration is taken as the research object to study the heat transfer of tooth surface and the heat distribution of tooth body temperature field under different working conditions. In the second section, the three-dimensional model of small module gear with curve configuration is designed and established. The third section analyzes the motion speed, normal load, and contact stress of the gear at the meshing point, and prepares for the calculation of the friction heat flux in the fourth section. Section “Thermal analysis of gear pair” studies the steady-state temperature field and transient temperature field under different working conditions. The conclusion is put forward in section “Conclusions.” The flowchart of the model implementation is shown in Figure 1.

Figure 1.

Flowchart of the model implementation.

Design method of small module gear with curve configuration

Normal tooth profiles design

A two-point contact small module gear with curve configuration is designed based on the previous study.³⁰ The normal tooth profile of the designed small module gear with curve configuration is shown in Figure 2, and the basic parameters of the tooth profile are shown in Table 1. In Figure 2(b), C₁ and C₂ are two contact points, namely, two meshing points of convex and concave teeth.

Figure 2.

Normal tooth profiles: (a) the convex pinion shape and (b) the concave tooth space.

Table 1.

Basic design requirements for the generated tooth profiles.

Parameters	The pinion	The gear
Tooth angle at contact points α₁, α₂	18°, 40°	18°, 40°
Tooth height h₁, h₂	1.5m_n	1.52m_n
Tooth addendum h_a₁, h_a₂	1.234m_n	0.166m_n
Tooth dedendum h_f₁, h_f₂	0.266m_n	1.354m_n
Circular arc radius ρ_a, ρ_f	1.5m_n	1.65m_n
Circular displacement e_a, e_f	0	0.0634m_n
Circular center offset l_a, l_f	0.5895m_n	0.5595m_n
Tooth thickness S_a, W_f	1.54m_n	1.5416m_n
Space width w_ak, w_fk	1.6016m_n	1.60m_n
Tooth flank gap j	/	0.06m_n
Tooth root radius ρ_ea, ρ_ef	0.4m_n	0.452m_n

m_n is the module.

As shown in Figure 2(a), the normal tooth profile of the convex tooth is composed of two arc lines M₁M₂ and M₂M₃. The parameter equations in the coordinate system O_n-X_nY_n are shown in equations (1) and (2). In the following parameter equation, “+” represents the left tooth surface of the gear tooth, and “−” represents the right tooth surface of the gear tooth.

r_{F}^{(I)} (θ_{ea}) = [\begin{matrix} \pm [ρ_{ea} \sin α_{min} - (ρ_{a} + ρ_{ea}) \cos α_{min} + l_{a}] \\ - ρ_{ea} \cos θ_{ea} + (ρ_{a} + ρ_{ea}) \sin α_{min} \\ 0 \\ 1 \end{matrix}]

(1)

In equation (1), $ρ_{ea}$ is the radius of M₁M₂ segment of arc; $l_{a}$ denotes the distance from the origin O₀ to the Y_n axis; $θ_{ea}$ is a variable parameter that determines the position of the upper point of the tooth root arc M₁M_2; $ρ_{a}$ is the radius of the arc M₂M₃ segment; $α_{min}$ is the minimum value of design parameter $α_{a}$ .

r_{F}^{(II)} (α_{a}) = [\begin{matrix} \mp (ρ_{a} \cos α_{a} - l_{a}) \\ ρ_{a} \sin α_{a} \\ 0 \\ 1 \end{matrix}]

(2)

In equation (2), $α_{a} (α_{min} \leq α_{a} \leq α_{max})$ is the variable parameter that determines the position of the point on the working arc M₂M₃.

As shown in Figure 2(b), the normal tooth profile of the concave tooth is composed of four curves, D₁D₂ is the chamfering line of the tooth top, D₂C₁ is the transition arc with a radius of $ρ_{f 1}$ , C₁D₃ is a parabola, and D₃D₄ is the tooth top arc with a radius of $ρ_{ef}$ . The parametric equation of the four-segment curve in the coordinate system O_n-X_nY_n is shown in equations (3)–(6).

r_{P}^{(II)} (m) = [\begin{matrix} \mp [m \sin δ_{e} + ρ_{f 1} \cos α_{0} - (ρ_{f 1} - ρ_{a}) \cos α_{1} - l_{f}] \\ ρ_{f 1} \sin α_{0} - (ρ_{f 1} - ρ_{a}) \sin α_{1} - m \cos δ_{e} \\ 0 \\ 1 \end{matrix}]

(3)

In equation (3), m denotes the distance from the point D₂ to any point on the line D₁D_2; $l_{f}$ is the eccentricity of the center of the circle; $α_{0}$ represents the angle between the normal line and the nodal line of point D_2; $δ_{e}$ is the angle between D₁D₂ and Y_n axis.

r_{P}^{(II)} (α_{f}) = [\begin{matrix} \mp [ρ_{f 1} \cos α_{f} - (ρ_{f 1} - ρ_{a}) \cos α_{1} - l_{f}] \\ ρ_{f 1} \sin α_{f} - (ρ_{f 1} - ρ_{a}) \sin α_{1} \\ 0 \\ 1 \end{matrix}]

(4)

In equation (4), $α_{f}$ is the parameter determining the position of the point in the arc D₂C_1; $ρ_{f 1}$ is the radius of arc D₂C₁.

r_{P}^{(III)} (t) = [\begin{matrix} \pm (t \sin α_{f 12} + \frac{t^{2}}{2 p_{1}} \cos α_{f 12} - L_{1} \cos α_{f 12} + l_{f}) \\ t \cos α_{f 12} - \frac{t^{2}}{2 p_{1}} \sin α_{f 12} + L_{1} \sin α_{f 12} \\ 0 \\ 1 \end{matrix}]

(5)

In equation (5), $t \in [- ρ_{a} \sin θ_{1}, ρ_{a} \sin θ_{1}]$ is the variable parameter of the parabola C₁D₃, $θ_{1} = (α_{2} - α_{1}) / 2$ , $α_{f 12} = (α_{1} + α_{2}) / 2$ ; $p_{1} = ρ_{a} \cos θ_{1}$ is the parabolic coefficient; $L_{1} = ρ_{a} \sin^{2} θ_{1} / 2 \cos θ_{1} + ρ_{a} \cos θ_{1}$ is the distance from the vertex of the parabola to the origin O_n.

r_{P}^{(IV)} (θ_{ef}) = [\begin{matrix} \mp ρ_{ef} \sin θ_{ef} \\ ρ_{ef} (\cos θ_{ef} - 1) + h_{f 2} \\ 0 \\ 1 \end{matrix}]

(6)

In equation (6), $θ_{ef}$ is the variable parameter that determines the position of the point on the arc D₃D_4; $ρ_{ef}$ is the radius of arc D₃D_4; $h_{f 2}$ is the tooth root height.

Establishment of 3D gear model

According to the Liang et al.,³¹ the tooth surface equation of the gear is obtained by using the rack cutter. The tooth surface point data set can be calculated from the tooth surface equation, and the three-dimensional model of the gear pair is established by means of the modeling software, as shown in Figure 3. The parameter design of gear pair is shown in Table 2.

Figure 3.

Gear pair model.

Table 2.

Basic design parameters of gear pair.

Parameters	Values
Module m_n (mm)	0.5
Tooth number of pinion z₁	11
Tooth number of wheel z₂	38
Helix angle β (°)	24.105
Tooth width B (mm)	5
Total contact ratio ε	1.3
Center distance a (mm)	13.420

Contact analysis

Gear with curve configuration is a new type of gear. In order to study the friction heat generation, the tangential velocity, relative sliding velocity, and load distribution of the meshing point are derived in this section. It provides a basis for calculating the frictional heat flux density in the fourth section.

Motion velocity analysis of contact point

Gear with curve configuration transmission is to push the teeth of the driven wheel along the axial direction by the teeth of the driving wheel, so as to realize the transmission of motion and power, and its contact point moves along the axial direction. Since the motion characteristics of the two contact points are similar, the motion analysis of the first contact point C₁ is taken as an example to derive its motion speed. In Figure 4, A₁₁j₁₂ and B₁₁j₁₂ are the two contact cylindrical helices of the first pair of contact traces. j₁₁j₁₂ parallel to the rotation z-axis is the first line of engagement. A₁j₁₂ and B₁j₁₂ are tangents of the contact lines A₁₁j₁₂ and B₁₁j₁₂, respectively.

Figure 4.

Contact point C₁ velocity analysis.

The motion behavior of contact point C₁ is analyzed by vector graphic method. As shown in Figure 4, the cylindrical tangents A₁j₁₁ and B₁j₁₁ represent the circumferential velocities V_B₁₁ and V_B₁₂ of the large and small gears at the contact point C₁. The helical tangents A₁j₁₂ and B₁j₁₂ represent the contact point C₁ linear velocities V_j₁₁ and V_j₁₂. From the vector relationship, the linear velocity V_j₁₁ and V_j₁₂ of the meshing point C₁ moving along the contact line are shown in Formula (7).

{\begin{matrix} V_{j 11} = \frac{V_{B 11}}{\sin β_{B 11}} \\ V_{j 12} = \frac{V_{B 12}}{\sin β_{B 12}} \end{matrix}

(7)

Where V_B₁₁ and V_B₁₂ are the circumferential velocity of the gear, and β_B₁₁ and β_B₁₂ are the helical angles.

The relative sliding velocity vector decomposition of the contact point C₁(j₁₁) is shown in Figure 5. Each component is obtained as shown in formula (8).

{\begin{matrix} V_{slip 1 x} = - (ω_{1} + ω_{2}) l_{1} \sin α_{1} \\ V_{slip 1 y} = (ω_{1} + ω_{2}) l_{1} \cos α_{1} \end{matrix}

(8)

Figure 5.

Vector decomposition of relative sliding velocity V_slip₁ at contact point C_1.

Thus, the magnitude and direction of the relative sliding velocity V_slip₁ are obtained, as shown in formula (9).

{\begin{matrix} V_{slip 1} = \sqrt{V_{slip 1 x}^{2} + V_{slip 1 y}^{2}} = l_{1} (ω_{1} + ω_{2}) \\ \frac{V_{slip 1 x}}{V_{slip 1 y}} = \frac{- (ω_{1} + ω_{2}) l_{1} \sin α_{1}}{(ω_{1} + ω_{2}) l_{1} \cos α_{1}} = - \tan α_{1} \end{matrix}

(9)

Where $α_{1}$ is the pressure angle of contact point C₁. It can be concluded that the relative sliding velocity V_slip₁ at C₁ is perpendicular to the offset pj₁₁.The magnitude and direction of the relative sliding velocity V_slip₂ at the second contact point C₂ can also be obtained, as shown in formula (10).

{\begin{matrix} V_{slip 2} = \sqrt{V_{slip 2 x}^{2} + V_{slip 2 y}^{2}} = l_{2} (ω_{1} + ω_{2}) \\ \frac{V_{slip 2 x}}{V_{slip 2 y}} = \frac{- (ω_{1} + ω_{2}) l_{2} \sin α_{2}}{(ω_{1} + ω_{2}) l_{2} \cos α_{2}} = - \tan α_{2} \end{matrix}

(10)

Where $α_{2}$ is the pressure angle of contact point C₂. Because l₁ and l₂ are the radius ρ_a of the working tooth profile of the convex tooth. From the formula (9) and (10), it can be concluded that the relative sliding speed at the two meshing points is equal in different directions, and the relative sliding speed is proportional to the gear module.

Normal load at contact point

As shown in Figure 6, the circular arc is tangent to the parabola at C₁ and C₂, while C₁ and C₂ are the contact points. Since the center of the circular arc profile is on the pitch line of the gear, both O₀C₁ and O₀C₂ are normal vectors of the contact points of the circular arc profile and the parabola profile. The normal load at the two contact points is shown in formula (11).

{\begin{matrix} F_{t} = \frac{T}{d} \\ F_{n} = \frac{F_{t}}{\cos β \cdot \cos α} \\ F_{n 1} = F_{n 2} = \frac{F_{n}}{2 \cos (\frac{α_{2} - α_{1}}{2})} \end{matrix}

(11)

Where $F_{t}$ is the circumferential force, $T$ is the load moment, $d$ is the radius of the indexing circle, $F_{n}$ is the total normal load, $F_{n 1}$ is the C₁ normal load at the contact point 1, and $F_{n 2}$ is the C₂ normal load at the contact point 2.

Figure 6.

Normal load decomposition.

Contact stress analysis

Since the small module gear with curve configuration is a typical point contact, the contact stress solution method adopts the Hertz point contact theory. As shown in Figure 7, the Hertz point contact is deformed into an elliptical region.

{\begin{matrix} a = m_{a} \sqrt[3]{\frac{3 F_{n}}{2 E \sum ρ}} \\ b = m_{b} \sqrt[3]{\frac{3 F_{n}}{2 E \sum ρ}} \\ S = m_{a} m_{b} π \sqrt[3]{{(\frac{3 F_{n}}{2 E \sum ρ})}^{2}} \end{matrix}

(12)

σ_{max} = \frac{3 F_{n}}{2 π ab}

(13)

Where $σ_{max}$ is the maximum contact stress. The parameters $m_{a} = \sqrt[3]{2 L (e) / π k^{2}}$ and $m_{b} = \sqrt[3]{2 L (e) k / π}$ characterize the semi-axis coefficients of the contact ellipse, and $k = b / a$ denotes the elliptic rate. The variable $e = \sqrt{1 - k^{2}}$ is used to express the elliptic eccentricity. $G (e) = \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{1 - e^{2} \sin^{2} φ}} d φ$ and $L (e) = \int_{0}^{\frac{π}{2}} \sqrt{1 - e^{2} \sin^{2} φ} d φ$ are associated with the first and second types of fully elliptic integrals. The symbol $E = 1 / ((1 - v_{1}^{2}) / E_{1} + (1 - v_{2}^{2}) / E_{2})$ is employed for the equivalent module of elasticity, while $E_{1}$ and $E_{2}$ refer to the material module of elasticity. Furthermore, $v_{1}$ and $v_{2}$ denote Poisson’s ratio, and $\sum ρ$ represents the algebraic sum of principal curvatures.

Figure 7.

Hertz point contact model.

From the material parameters in Table 3, the contact stress and contact deformation of each contact area are obtained by numerical calculation, as shown in Figures 8 and 9. Under the load of 160 N mm, the contact stress of the first contact point C₁ in the single tooth meshing area is 202.14 MPa, and the contact stress in the double tooth meshing area is 160.44 MPa. The contact stress of the second contact point C₂ is 124.84 MPa in the single-tooth meshing area and 99.09 MPa in the double-tooth meshing area. The contact stress in the single-tooth meshing area is higher than that in the double-tooth meshing area. The contact pressure of the first contact point C₁ of the curve configuration gear is larger than that of the second contact point C₂. Excessive contact stress may lead to local overload, which will increase the risk of fatigue cracks and reduce fatigue life. The increase of temperature usually reduces the strength and hardness of the material, resulting in a shortened fatigue life. Considering the influence of temperature rise, according to the maximum stress criterion of fatigue failure, the contact stress of the single tooth contact area of the first contact point C₁ can be used as a criterion to judge whether the curve configuration gear is invalid.

Table 3.

Material parameters of gear pair.

Parameters	Values (23°C)
Coefficient of thermal expansion λ (1/°C)	2.1 × 10⁻⁵
Density ρ (kg/m³)	8.4 × 10³
Thermal conductivity k (W/(m °C))	125.1
Poisson’s ratio μ	0.346
Elastic module E (GPa)	106
Specific heat capacity C (J/(kg °C))	377

Figure 8.

Hertzian contact stress at the pinion torque of 160 N mm: (a) single tooth contact stress at first contact point C₁, (b) double-tooth contact stress at first contact point C₁, (c) single tooth contact stress at second contact point C₂, and (d) double-tooth contact stress at second contact point C₂.

Figure 9.

Contact stress at different torques: (a) single tooth contact stress at first contact point C₁, (b) double-tooth contact stress at first contact point C₁, (c) single tooth contact stress at second contact point C₂, and (d) double-tooth contact stress at second contact point C_2.

Thermal analysis of gear pair

Boundary conditions

The contact line of gear meshing transmission with curve configuration. At any time, two points contact with each other. In order to intuitively simulate the temperature field distribution during gear meshing, based on the finite element model, the heat flux of the Hertz contact elliptical heat source is calculated and applied to the spiral contact trace. At the same time, each tooth surface conducts convective heat transfer with dry air, as shown in Figure 10. In this model, the heat flux entering the gear is evenly distributed on the contact trace of the gear surface, and the heat flux at any position in the interval is obtained by formula (23) and formula (24). The spiral bandwidth is the Hertz point contact width 2b. In the case of dry contact, it is assumed that the heat source is generated by the friction of two contact points, and the temperature of the surrounding medium is the same as that of the air.²³ The convective heat transfer coefficient of each tooth is expressed by the equations (18)–(20).

Figure 10.

Simulation of friction heat and convective heat transfer diagram.

As shown in Figure 11, region 1 is the meshing surface of convex and concave teeth, and its boundary conditions are shown in equation (14).

- λ \frac{\partial T}{\partial n} = h_{a} (T - T_{0}) - Q

(14)

Where n represents the direction perpendicular to the surface of region 1, h_a represents the convective heat transfer coefficient of the surface of region 1, and Q represents the frictional heat input of region 1.

Figure 11.

Gear convection heat transfer model.

Region 2 is the end face and its boundary conditions are shown in equation (15).

- λ \frac{\partial T}{\partial n} = h_{b} (T - T_{0})

(15)

Where $h_{b}$ is the convective heat transfer coefficient at the surface of region 2.

Region 3 is the non-meshing surface such as gear tooth root surface and transition surface, and its boundary conditions are shown in equation (16).

- λ \frac{\partial T}{\partial n} = h_{c} (T - T_{0})

(16)

Where $h_{c}$ is the convective heat transfer coefficient at the surface of region 3.

Heat transfer coefficients

The convective heat transfer state on the tooth surface of small module gear with curve configuration can be simplified as convective heat transfer on a rotating disk. The medium Reynolds number (Re) is calculated as follows.

Re = \frac{ω r^{2}}{v_{f}}

(17)

Where $ω$ is the angular velocity, in rad/s. $r$ is the radius of rotation, in mm. $v_{f}$ is the kinematic viscosity, in mm²/s.

The heat transfer coefficients h_a for the mesh face flow is as follows.

h_{a} = \frac{0.228 {Re}^{0.731} P r^{\frac{1}{3}} k_{L}}{d_{i}}

(18)

Where $\Pr = ρ_{L} C_{L} v_{f} / k_{L}$ is the medium Prandtl number. $ρ_{L}$ is the medium density, in kg/m³. $C_{L}$ is the medium specific heat capacity, in J/(kg °C). $k_{L}$ is the medium coefficient of thermal conductivity, in W/(m °C). $d_{i} (i = 1, 2)$ is the size of the gear diameter of the indexing circle, in mm.

After calculation, the Reynolds number is less than 2 × 10⁵ in the working range of the study, which indicates that the fluid on the gear end face is laminar flow, and the convective heat transfer coefficient h_b is shown in formula (19).

h_{b} = 0.308 k_{L} {(m_{h} + 2)}^{0.5} \Pr^{0.5} {(\frac{ω_{i}}{v_{f}})}^{0.5}

(19)

Where $m_{h}$ represents the calculated heat transfer coefficient factor, taken as 2. $ω_{i} (i = 1, 2)$ denotes the rotational speed of the gear and the pinion, respectively, in rad/s.

The flow heat transfer coefficients $h_{c}$ for non-meshing surfaces such as root surfaces and transition surfaces is as follows.

h_{c} = 0.664 k_{L} \overset{\frac{1}{3}}{\Pr} {(\frac{ω_{i}}{v_{f}})}^{0.5}

(20)

In order to simulate the temperature rise of gear with curve configuration under dry conditions, dry air with a pressure of 100 kPa at 23°C is selected as the medium parameter. The heat transfer coefficient of each tooth surface at different rotational speeds is calculated by the formulas (17)–(20) as shown in Figure 12.

Figure 12.

Convective heat transfer coefficient.

Friction heat flux

During a meshing cycle, the gear generates heat due to friction transfer motion and power during meshing. After exiting the meshing, the temperature of the gear teeth is reduced due to thermal convection and heat conduction with the environment. After a long time of work, the gear and the surrounding environment reach a thermal equilibrium state, that is, the temperature field distribution of the gear tooth is stable when it is not meshed. Only when it enters the meshing again, the temperature of the gear tooth will increase to reach the maximum meshing flash temperature.

The instantaneous frictional heat power of the gear on the unit contact line is shown in equation (23).¹⁰

Q_{S} = β_{k} F_{n} u V_{slip}

(21)

β_{k} = \frac{\sqrt{k_{1} ρ_{1} C_{1} V_{j 1 i}}}{\sqrt{k_{1} ρ_{1} C_{1} V_{j 1 i}} + \sqrt{k_{2} ρ_{2} C_{2} V_{j 2 i}}}

(22)

Where $F_{n}$ is the contact point normal load; β_k is the heat distribution coefficient; u = 0.35 is the sliding friction coefficient; V_slip is the relative sliding velocity.

The instantaneous frictional heat flow density $q_{i}$ of the helical tooth surface per unit contact line is as follows.

q_{i} = β_{k} σ_{a 1 i} u V_{slip}

(23)

In order to obtain the steady-state temperature field of gear, the instantaneous thermal load of meshing is averaged in a meshing period. The calculation equation of the average frictional heat flux of each contact points in a meshing cycle is shown in equation (21).²³

{\begin{matrix} Q_{1} = \frac{γ_{k} β_{k} u σ_{max} a ω_{1} V_{slip}}{π V_{j}} \\ Q_{2} = \frac{γ_{k} (1 - β_{k}) u σ_{max} a ω_{2} V_{slip}}{π V_{j}} \end{matrix}

(24)

In the equations, γ_k represents the coefficient of friction converted into heat, usually between 0.9 and 0.95.²³Q₁ and Q₂ are the meshing friction heat flux of convex and concave teeth, respectively. σ_max is the maximum Hertz contact stress; a is Hertz contact half width; ω₁ and ω₂ are the rotational speed of the convex-concave gear respectively; V_j is the tangential velocity. Taking equations (7)–(13) into equation (24), the average frictional heat flux can be obtained as shown in Figures 13 and 14.

Figure 13.

Friction heat flux under load of 160 N mm.

Figure 14.

Friction heat flow of pinion at 2000 rpm.

Thermal analysis of gear mesh

Pre-processing of the simulation

Because the curve configuration gear is a spiral structure convex and concave tooth type, the ANSYS mesh automatic division function is difficult to achieve the ideal mesh of the model. In this paper, the gear contact ratio is designed to be 1.3. In order to apply heat flux to the single and double tooth meshing area, the number of grids along the tooth width direction of the convex and concave teeth is guaranteed to be the same, so use HyperMesh to manually divide the grid.

In order to ensure that the finite element mesh model can obtain the accurate solution in mathematics, the mesh convergence is studied. Different mesh sizes are divided, and the element mesh type is Solid278. The mesh convergence analysis is carried out by simulating the dry operating conditions, the speed of the convex tooth is 2000 rpm and the torque of the concave tooth is 160 N mm. The ambient temperature is 23°C. The material parameters of brass varying with temperature are set in the simulation software, and the material parameters at 23°C are shown in Table 3. The grid convergence curve is shown in Figure 15, and the steady-state temperature field of different grid sizes is shown in Figure 16. As shown in Figures 15 and 16, the latter three sets of grid model grids basically give the same results, so it can be considered that the model has obtained an accurate mathematical solution. The difference between the third group and the fifth group of simulation data is the smallest. Therefore, the number of convex tooth mesh model nodes is 12,614, the number of elements is 10,400, the number of concave tooth model nodes is 13,727, the number of elements is 11,648, and the temperature field simulation analysis is carried out.

Figure 15.

Temperature field diagrams of different mesh sizes.

Figure 16.

Temperature field diagrams of different mesh sizes: (a) 6993 nodes and 5824 elements of concave tooth, (b) 6426 nodes and 5200 elements of convex teeth, (c) 8029 nodes and 5824 elements of concave tooth, (d) 7378 nodes and 6000 elements of convex teeth, (e) 10,360 nodes and 8736 elements of concave tooth, (f) 9520 nodes and 7800 elements of convex teeth, (g) 10,618 nodes and 8960 elements of concave tooth, (h) 9758 nodes and 8000 elements of convex teeth, (i) 13,727 nodes and 11,648 elements of concave tooth, and (j) 12,614 nodes and 10,400 elements of convex teeth.

Steady-state temperature field analysis

In this section, the steady-state temperature rise law of small module gear with curve configuration in dry operation under different speed and torque conditions is explored. The ambient temperature is 23°C and the average friction heat flow, convective heat transfer coefficient h_a, end face heat transfer coefficient h_b, and non-meshing face heat transfer coefficient h_c calculated by the above formulas are input into ANSYS for steady-state temperature field simulation.

Figure 17 shows the heat distribution map of the finite element results of the small modulus curve configuration gear pair under the dry operating condition of 160 N mm load and 600 rpm speed of the convex gear. The maximum steady-state temperature is distributed at the first contact point C₁ in the middle of the single-tooth meshing area. The temperature range of the convex gear tooth body is 61.94°C–62°C, and the temperature range of the concave gear tooth body is 53.624°C–53.646°C. It shows that the temperature distribution on the tooth surface of the small modulus curve configuration gear pair is not much different after reaching the heat balance. The reason is that the size of the small modulus gear pair is small, and the heat generated by friction is transmitted to the whole tooth body quickly. Therefore, after the thermal balance with the environment, the temperature difference between each part of the tooth body is very small.

Figure 17.

Steady-state temperature field diagram: (a) pinion steady-state temperature field and (b) large gear steady-state temperature field.

As shown in Figure 18, there is an approximate linear growth relationship between gear speed and gear meshing temperature rise. This is due to the increase of meshing times per unit time, resulting in a large temperature rise. In addition, the number of meshing times per unit time of the convex tooth is more than that of the concave tooth, resulting in the temperature rise of the convex tooth at different speeds higher than that of the concave tooth. Table 4 shows that the temperature difference between the convex and concave teeth increases with the increase of the rotational speed under a certain torque.

Figure 18.

Gear temperature at different speeds under load of 160 N mm.

Table 4.

The steady-state temperature difference of the gears varies with the speed.

Pinion rotation speed (rpm)	The difference of steady-state temperature field between convex gear and concave gear (°C)
600	8.355
800	9.555
1000	10.638
1200	11.62
1400	12.592
1600	13.457
1800	14.249
2000	15.981

As shown in Figure 19, the gear torque and the gear meshing temperature rise show an approximately linear increasing relationship. The reason is that as the torque increases, the load force borne by the gear increases, resulting in an increase in friction heat. It can be seen from Table 5 that the temperature difference between the convex tooth and the concave tooth increases with the increase of the load at a certain speed.

Figure 19.

Gear temperature under different torques.

Table 5.

The steady-state temperature difference of the gears varies with the torque.

Wheel torque (N mm)	The difference of steady-state temperature field between convex gear and concave gear (°C)
20	4.744
40	6.923
60	8.837
80	10.429
100	11.966
120	13.394
140	14.712
160	15.981

Further, the meshing temperature test rig for the small module gear with curve configuration is constructed in Figure 20. Here, the infrared thermal imager is set to automatically capture the maximum temperature. The measurement position is 3 cm above the gearbox and the temperature is measured each 2000 revolutions. The temperature value obtained from each measurement is calibrated. If the maximum measured temperature obtained from subsequent tests is less than 2°C compared with the absolute value of the difference between the calibrated data for five consecutive times, it means that the gear pair reaches the temperature stable operation period. The last three measured values of each group are taken as the results.

Figure 20.

Temperature measuring test bench. 1. Drive motor, 2. Output shaft coupling, 3. Motor speedometer, 4. Gearbox, 5. Miniature magnetic powder brake, 6. Infrared thermal imager, 7. Driven wheel concave teeth, 8. Driving wheel convex teeth.

The flash temperature will dissipate significantly within a short time dt when the gear pair separates from contact point.³² The rotation direction of gear and temperature rise of tooth surfaces during the meshing process are shown in Figure 21(a). The maximum temperature happens at the meshing-out point of the convex tooth. At this time, the flash temperature dissipates after half a meshing period. Figure 21(b) shows the thermal image of the gear pair during the temperature stability stage under different working conditions.

Figure 21.

Maximum steady-state temperature test value: (a) gear temperature rise change rule and (b) infrared thermal image under different working conditions.

Through the analysis results in Figure 22, the error values between simulation analysis and experimental test are 4.84%, 3.59%, 6.16%, and 7.10% under the torque of 40, 80, 120, and 160 N mm, respectively. Variation trend of the maximum steady-state temperature is in good agreement with those of experimental test, and the maximum error is less than 7.1%.

Figure 22.

Results comparisons of the maximum steady-state temperature.

Simulation of transient temperature field of gear tooth

Because the steady-state temperature field of the convex tooth is greater than that of the concave gear under different working conditions, the flash temperature analysis of the convex tooth is carried out. When analyzing the transient temperature field, the instantaneous frictional heat flux is evenly divided into a certain number of load steps along the two contact tracks of the gear to simulate the thermal load motion. After each loading step is completed, the heat flow q_i advances one unit in the pinion meshing direction and removes the heat flow from the previous loading step. In addition, the calculation result of the previous loading step is used as the initial condition of the next loading step. In order to obtain the flash temperature of the tooth surface, the obtained steady-state temperature field of the convex tooth is first used as the initial condition of the transient thermal analysis, and then transferred to the transient thermal analysis module. According to the solution time and time increment, the ANSYS APDL code is written by the cyclic command to realize the continuous step-by-step loading of the transient friction thermal load at the meshing point, and the transient thermal analysis results are obtained. The simulation process is shown in Figure 23.

Figure 23.

Flow chart of transient temperature simulation.

Figure 24 shows the transient temperature field distribution at different times under the dry operation of the convex tooth speed at 2000 rpm and the load of 160 N mm. The maximum flash temperature occurs in the single-tooth meshing area of the first contact point C₁, which is 102.44°C.

Figure 24.

Transient temperature field at different time: (a) double-toothed meshing, (b) single tooth meshing, and (c) double-toothed meshing.

The ISO standard flash temperature method is based on the maximum meshing temperature T_t on the tooth surface of the tooth surface to determine whether the tooth surface is gluing failure. The maximum transient temperature T_t consists of the steady-state temperature T_m and the tooth surface flash temperature T_flash, as shown in Formula (25).

T_{t} = T_{m} + T_{flash}

(25)

The ISO standard Blok flash temperature formula for gears with strip contact and parallel tangential velocities is shown in Formula (26).³³

T_{flash} = 1.11 \frac{u X_{Γ} X_{J} W_{Bn} (| V_{1} - V_{2} |)}{\sqrt{2 a} (B_{m 1} \sqrt{V_{1}} + B_{m 2} \sqrt{V_{2}})}

(26)

Where $u$ is the friction factor. $X_{Γ}$ is the load distribution coefficient. $X_{J}$ is the engagement coefficient, taking 1 for the deceleration device. $W_{Bn}$ is the unit load at the point of contact. $V_{1}$ , $V_{2}$ are the tangential speeds of the master and follower wheels respectively. $a$ is the Hertzian contact half-width. $B_{mi} = \sqrt{0.001 \cdot k_{i} ρ_{i} C_{i}}$ is the thermal contact coefficient.

The Blok theoretical value of the maximum meshing temperature of the small modulus curve configuration gear under four different torque dry friction conditions is compared with the finite element simulation value, as shown in Figure 25. The flash temperature of Blok theory is higher than that of finite element simulation, and the difference is within 5°C. Because the Blok theory only considers the heat conduction in the direction perpendicular to the tooth surface, the predicted maximum flash temperature is higher than the true value.

Figure 25.

The maximum flash temperature of Blok theory and finite element simulation.

The increase of temperature will soften the material and easily lead to scratches on the surface of the gear. And the increase of temperature leads to the decrease of material strength. When the gear surface is subjected to large contact stress, plastic deformation or strain may occur, which affects the normal meshing of the curve configuration gear. The thermal expansion caused by high temperature leads to the change of gear meshing clearance, which makes the meshing accuracy lower and the transmission stability worse. Therefore, according to the working temperature and considering the thermal expansion property of the material, the design clearance of the tooth profile of the curve configuration gear should be adjusted. And the thermal cycle under high temperature conditions will accelerate the fatigue of the material, thermal fatigue may lead to cracks and spalling on the surface of the gear. The maximum flash temperature of the curve configuration gear occurs in the single tooth meshing area of the first contact trace, which may be the first to occur gluing failure. Therefore, in the design of curve configuration gear, the contact ratio should be increased as much as possible, the number of meshing teeth should be increased, and the load should be shared to achieve the purpose of reducing the temperature rise. In addition, proper heat dissipation and lubrication are also important.

Conclusions

(1) A small module gear with curve configuration is proposed considering the special tooth form with circular-arc and parabola curves. The normal tooth profile of the gear pair is designed and tooth surfaces equations are deduced based on the principle of gear meshing. The accurate three-dimensional models of the gear pair are established.

(2) Theoretical contact analysis between mated tooth surfaces including the motion velocity, normal load, and contact stress at the contact point are carried out. The relative sliding velocities at contact points are equal. The contact stresses and contact deformations at contact points in each meshing region are obtained. It is observed that the contact pressure at the first meshing angle on contact point C₁ is larger than that at the second meshing angle on contact point C₂.

(3) The single tooth mesh model of the gear pair is established using HyperMesh, and the single-double-tooth meshing region of two contact points is delineated. Meshing model of thermal analysis of the gear pair based on finite element method is established under different working conditions. The temperature rise of the gear sub-mesh increases with high rotational speed and torque. Under the same working conditions, the pinion exhibits greater frictional heat generation than the gear. The differences in the heat distribution of the steady-state temperature field of the small module gear are minimal.

(4) The maximum temperature distribution locates at the first meshing angle of contact point in the middle of the single tooth meshing region under different working conditions. Based on the steady state temperature field results, the transient temperature field results are obtained by discretizing the heat source load motion through ANSYS APDL parametric programming.

(5) Meshing temperature test rig for the small module gear with curve configuration is constructed. The maximum steady-state temperature simulation value is in good agreement with the variation trend of the test value, and the maximum error is less than 7.1%. The simulation results of flash temperature also agree well with Blok theory. The method can also be extended to other point contact gear pair.

Footnotes

Handling Editor: Fatih Karpat

Author contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Dong Liang, Yun Liu, Qunlong Sun, and Chengli Hua. The typesetting and proofreading format were completed by Qunlong Sun and Chengli Hua. Supervision by Dong Liang. The first draft of the manuscript was written by Yun Liu and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work was supported by National Natural Science Foundation of China (Grant No. 52175042), Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJZD-M202400705), Graduate Research Innovation Project of Chongqing City (Grant No. CYS240488), and Research Project of Chongqing College of Electronic Engineering (Grant No. XJZK-202103).

Consent to publication

All authors have given consent to publish this manuscript.

ORCID iD

Dong Liang

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Meshing model and temperature field analysis of small module gear with curve configuration

Abstract

Keywords

Introduction

Design method of small module gear with curve configuration

Normal tooth profiles design

Establishment of 3D gear model

Contact analysis

Motion velocity analysis of contact point

Normal load at contact point

Contact stress analysis

Thermal analysis of gear pair

Boundary conditions

Heat transfer coefficients

Friction heat flux

Thermal analysis of gear mesh

Pre-processing of the simulation

Steady-state temperature field analysis

Simulation of transient temperature field of gear tooth

Conclusions

Footnotes

Author contributions

Declaration of conflicting interests

Funding

Consent to publication

ORCID iD

References